A homogeneous polynomial is called completely reducible over a field \(\mathbb {F}\) if it can be factored into a product of \(\mathbb {F}\) -linear forms. Set \(\mathbb {F}=\mathbb {R}\) , \(r^{2}=x^{2}+y^{2}+z^{2}\) , and let \(\mathcal {H}_{n}\) and \(\mathcal {D}_{n}\) be the families of all harmonic and all completely reducible polynomials in the space \(\mathcal {P}_{n}(\mathbb {R}^{3})\) of homogeneous polynomials of degree n, respectively. Then, for every \(h\in \mathcal {H}_{n}\) , there exist the unique \(p\in \mathcal {D}_{n}\) and \(q\in \mathcal {P}_{n-2}\) such that \(h=p+r^{2}q\) . Sylvester sketched a proof of this theorem in his note on spherical harmonics as a remark on Maxwell’s pole theory. If \(\mathbb {F}=\mathbb {C}\) , then the same equalities hold for some p, q, but they are usually not unique. For h in general position, there are \((2n-1)!!\) such polynomials. Let \({{\Lambda }}_{h}=(h+r^{2}\mathcal {P}_{n-2})\cap \mathcal {D}_{n}\) . We prove that \(h=\frac{1}{(2n-1)!!}\sum \limits _{p\in {{\Lambda }}_{h}}p\) , i.e., h is the center of mass of \({{\Lambda }}_{h}\) . This implies algebraic quadratic relations between the spherical harmonics.
